two-valued logic

logic (Commonly known as "Boolean algebra") A mathematical

system concerning the two truth values, TRUE and FALSE and

the functions AND, OR, NOT. Two-valued logic is one of

the cornerstones of logic and is also fundamental in the

design of digital electronics and programming languages.

The term "Boolean" is used here with its common meaning -

two-valued, though strictly Boolean algebra is more general

than this.

Boolean functions are usually represented by truth tables

where "0" represents "false" and "1" represents "true". E.g.:

A | B | A AND B

--+---+--------

0 | 0 | 0

0 | 1 | 0

1 | 0 | 0

1 | 1 | 1

This can be given more compactly using "x" to mean "don't

care" (either true or false):

A | B | A AND B

--+---+--------

0 | x | 0

x | 0 | 0

1 | 1 | 1

Similarly:

A | NOT A A | B | A OR B

--+------ --+---+--------

0 | 1 0 | 0 | 0

1 | 0 x | 1 | 1

1 | x | 1

Other functions such as XOR, NAND, NOR or functions of

more than two inputs can be constructed using combinations of

AND, OR, and NOT. AND and OR can be constructed from each

other using DeMorgan's Theorem:

A OR B = NOT ((NOT A) AND (NOT B))

A AND B = NOT ((NOT A) OR (NOT B))

In fact any Boolean function can be constructed using just NOR

or just NAND using the identities:

NOT A = A NOR A

A OR B = NOT (A NOR B)

and DeMorgan's Theorem.

(2003-06-18)